Exercise 7.5.7: A continuous : between metric spaces (, ) and (, ) is said to be proper if for every compact set , the set 1() is compact. Suppose a continuous : (0, 1) (0, 1) is proper and { }=1 is a sequence in (0, 1) converging to 0. Show that ( )=1 has no subsequence that converges in (0, 1)